Chapter 1 One-sample test of proportions

We will start by discussing the one-sample test of proportions. As an example, suppose it has been claimed that among social media users, 73% use Facebook more than once per day, and we wanted to test this claim. Consider the hypotheses

\[H_0 : p = 0.73 \text{ versus } H_1 : p \neq 0.73,\]

where:

  • \(p\) denotes the proportion of social media users who use Facebook more than once per day
  • \(H_0\) denotes the null hypothesis that the proportion of social media users who use Facebook more than once per day is equal to 0.73 (or as a percentage, 73%)
  • \(H_1\) denotes the alternative hypothesis that the proportion of social media users who use Facebook more than once per day is different from 73%.

In more general terms, suppose we have a random sample of \(n\) observations with an expected proportion \(p\) of these observations to have a certain characteristic, letting \(x\) denote the number of observations in the sample that actually have that characteristic. Equivalently, suppose we conduct \(n\) independent trials each with probability of success \(p\) and let \(x\) denote the number of successes in these \(n\) trials. Consider the hypotheses

\[H_0 : p = p_0\text{ versus } H_1 : p \neq p_0\text{ (or }p<p_0\text{ or }p>p_0),\]

where:

  • \(p_0\) denotes the expected proportion under the null hypothesis.

Then, provided \(n\) is not too small (this will be further discussed shortly), a commonly used statistical test for this type of hypothesis is the one-sample proportion test based on the estimate to \(p\), which we can denote as \(\hat{p} = x/n\).

Test your knowledge

In the example above where we are looking at the proportion of social media users who use Facebook more than once per day, what is the value of \(p_0\)?

0.73

Returning to our example, a survey was carried out (Raymond 2019) to study the social media habits of regular social media users from around the world. Supposing that of the \(n = 484\) respondents, \(x = 368\) said they used Facebook more than once per day, we then have that

\[\hat{p} = \frac{x}{n} = \frac{368}{484} \approx 0.76.\]

Once we know the values of \(x\) and \(n\), we then have enough information to calculate \(\hat{p}\) and then carry out the hypothesis test. Alternatively, if we know the value of \(n\) and \(\hat{p}\), we can use this information to calculate \(x\). Before carrying out the test, it is a good idea to visualise the data and check the assumptions, which we will do in the next section.

References

Raymond, Mark. 2019. “Social Media Usage Report 2019: User Habits You Need to Know.” 2019. https://www.goodfirms.co/resources/social-media-usage-user-habits-to-know.